2021/12/09 by Regeta, Andriy, van Santen, Immanuel
#14J50 #14M27 #22F50 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary: 14R20 #Secondary: 14M25
paper · doi:10.48550/arxiv.2112.04784
We describe maximal commutative unipotent subgroups of the automorphism group Aut(X) of an irreducible affine variety X. Further we show that a group isomorphism Aut(X) → Aut(Y) maps unipotent elements to unipotent elements, where Y is irreducible and affine. Using this result, we show that the automorphism group detects sphericity and the weight-monoid. As an application, we show that an affine toric variety different from an algebraic torus is determined by its automorphism group among normal irreducible affine varieties and we show that a smooth affine spherical variety different from an algebraic torus is determined by its automorphism group (up to an automorphism of the base field) among smooth irreducible affine varieties.